I do hope the trees are staying however having watched More London (I know, different estate) cut down a tree, do some work on cables under the ground where the tree was and then replant a new tree, I don't hold out much hope...
The 1799 map of Southwark (http://www.motco.com/map/81005/) shows that it's called St. John's Burying Ground and is attached to St. John's at the top of Fair Street/Church Row. It's also shown in the 1862 map (http://www.motco.com/map/81006/) I can't find a record for what happened to the park (and thus the bodies) after that, until the Ecology Park was established in the mid/late 70s? However there are a few gravestones left in Potters Field, leaning up against the wall on the Tooley Street end of the park, behind a bush.
I believe it's known as Potter's Field because one of the industries surrounding the cemetery was pottery, thus a lot of the buried were potters.
There was also a poor school in the park (St. John's Charity School).
There may also be some issue with St. Olave which was on the South side of London Bridge many years ago (roughly where St. Olave's House now is) and it may have been used as a cemetery for St. Olave's as well? In the map dated 1746 it's listed as St. Olave's Church Yard (http://www.motco.com/map/81002/), so perhaps later St. John's was established?